Consider the heat engine shown below. The cyclic integral of dQ is greater than zero. Does this violate the Clausius Inequality? Explain. |
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Read : |
This problem is
designed to make you think very carefully about how to use a cyclic integral. The key is that the temperature is not
the same for both of the heat transfer interactions in this cycle. |
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Diagram: |
See the problem
statement. |
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Given: |
TH |
400 |
K |
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QH |
450 |
kJ |
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TC |
300 |
K |
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QC |
-350 |
kJ |
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W |
-100 |
kJ |
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Find: |
Does the cycle violate the Clausius Inequality? |
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Assumptions: |
None. |
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Equations
/ Data / Solve: |
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The Clausius Inequality is: |
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Eqn 1 |
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Because the cycle only exchanges heat with the hot and cold thermal reservoirs, the integrals can be simplified: |
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Eqn 2 |
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Eqn 3 |
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100 |
kJ |
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-0.04167 |
kJ/K |
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This cycle is irreversible because the cyclic integral in the Clausius Inequality is less
than zero. |
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It is true that the cyclic integral of dQ > 0.
But the Clausius Inequality is still
satisfied. |
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Confusion about the cyclic integrals sometimes arises if you mistakenly pull T out of the cyclic integral. |
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Eqn 4 |
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You could only pull T out of the cyclic integral if ALL
of the heat exchange
across the system boundary went to
or from reservoirs that were ALL at the SAME temperature. |
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That is almost never going to happen. It is definitely not the case in this problem as the hot and cold reservoirs are at 400 K and 300 K,
respectively. |
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Verify: |
None. |
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Answers : |
Yes, the cycle does indeed satisfy the Clausius
Inequality. |
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